New Game Straight From Case to Console Disc Error?

I got this game recently, brand new, and took it straight from the case to my Xbox 360 S console. The console itself is only about 2 years old, and not heavily used. I handle the discs delicately so there is no damage or foreign objects that might obstruct it from reading. I get everything set up like signing into Uplay and such, and then go to play the single player campaign. The opening cutscene happens, then the naval combat, then the scene with Edward and Caroline, then it went to the game again to swim and it comes up with "Game Error This disc is unreadable. 1. Clean the disc with a soft cloth. 2. Restart the console." and so I thought maybe the console got overheated and turned it off and let it rest until it was cool. Then I turned it back on, started the game, and it would say "Game Error..." again while in the loading screen where you just run around as Edward. So I took out the disc, it is still fine, no scratches or anything, and put in a battered disc of Skyrim, which is a game I have issues with freezing some times. I played Skyrim for about 15 minutes with no issue whatsoever. I put Assassin's Creed IV back in to play. I decided to maybe start another game to see if that might help. Then it froze during the naval combat. I don't know what to do????